Difference between revisions of "Complex Numbers"

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(Created page with "<div class="noautonum">__TOC__</div> ==Introduction== The motivation for defining complex numbers comes from wanting to define <math> \sqrt{-1}</math> in a way that would be c...")
 
 
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==Introduction==
 
==Introduction==
 
The motivation for defining complex numbers comes from wanting to define <math> \sqrt{-1}</math> in a way that would be compatible with the properties of exponents.
 
The motivation for defining complex numbers comes from wanting to define <math> \sqrt{-1}</math> in a way that would be compatible with the properties of exponents.
−
Mathematicians accomplished this by defining a new "variable", i. Thus, we have the immediate property that <math> i^2 = -1 </math>, the same way <(\sqrt{3})^2 = 3 </math>.
+
Mathematicians accomplished this by defining a new "variable", i. Thus, we have the immediate property that <math> i^2 = -1 </math>, the same way <math>(\sqrt{3})^2 = 3 </math>.
 
Now we can define the Complex Numbers to be all numbers of the form <math> A + Bi </math> for real numbers <math> A </math> and <math> B</math>. The number A is called the real
 
Now we can define the Complex Numbers to be all numbers of the form <math> A + Bi </math> for real numbers <math> A </math> and <math> B</math>. The number A is called the real
 
part of <math> A + Bi</math>, while B is the imaginary part of <math> A + Bi</math>.
 
part of <math> A + Bi</math>, while B is the imaginary part of <math> A + Bi</math>.
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Given complex numbers z and w:
 
Given complex numbers z and w:
 +
 
1)<math> (\overline{\overline{z}}) = z </math>
 
1)<math> (\overline{\overline{z}}) = z </math>
 +
 
2) <math> \overline{z + w} = \overline{z} + \overline{w}</math>
 
2) <math> \overline{z + w} = \overline{z} + \overline{w}</math>
 +
 
3) <math> \overline{z\cdot w} = \overline{z}\overline{w} </math>
 
3) <math> \overline{z\cdot w} = \overline{z}\overline{w} </math>
 +
 +
 +
Now we are ready to find the inverse of any non-zero complex number A + Bi, that is find a complex number C + Di such that (A + Bi)(C + Di) = 1.
 +
 +
By the properties above, we know that <math>(A + Bi)(A - Bi) = A^2 + B^2</math> So given any complex number A + Bi, the inverse is <math> \frac{A - Bi}{A^2 + B^2}</math>. Another way to think of this is <math> \frac{1}{A + Bi} = \frac{A - Bi}{A^2 + B^2}</math>.
 +
 +
Example: Find the inverse of 3 + 4i.
 +
 +
First we find the value of <math> A^2 + B^2</math> for A = 3 and B = 4. So <math> A^2 + B^2 = 3^2 + 4^2 = 9 + 16 = 25</math>. Now the inverse of 3 + 4i is <math>\frac{3 - 4i}{25}</math>
 +
 +
 +
Another common question that requires knowledge of the inverse is to simplify an expression of the form <math> \frac{A + Bi}{C +Di}</math>.
 +
 +
Example:
 +
 +
Simplify <math>\frac{ 1 + 2i}{5 + 3i}</math>.
 +
 +
From the discussion about inverses above, we can rewrite this expression as <math> (1 + 2i)\cdot \frac{1}{5 + 3i}</math>, and the inverse of the second term is <math> \frac{5 - 3i}{ 5^2 + 3^2} = \frac{5 - 3i}{34}</math>. Putting this all together our expression has turned into <math> (1 + 2i)\cdot \frac{5 - 3i}{34}</math>. To finish the problem we just need to use FOIL and simplify. <math> ( 1 + 2i)\cdot \frac{5 - 3i}{34} = \frac{ 5 + 10i -3i -6i^2}{34} = \frac{11 + 7i}{34}</math>
  
 
==Powers if i==
 
==Powers if i==
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Some powers of i are provided here:
 
Some powers of i are provided here:
  
−
<math>i^1 = i
+
<math>i^1 = i</math>
  
−
i^2 = -1
+
<math>i^2 = -1</math>
  
−
i^3 = -i
+
<math>i^3 = -i</math>
  
−
i^4 = 1
+
<math>i^4 = 1
 
</math>.
 
</math>.
  
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1) If <math> b^2 - 4ac > 0</math> the equation <math> ax^2 + bx + c </math> has two distinct real roots.
 
1) If <math> b^2 - 4ac > 0</math> the equation <math> ax^2 + bx + c </math> has two distinct real roots.
 +
 
2) If <math> b^2 - 4ac = 0</math> the equation <math> ax^2 + bx + c </math> has a single real double root, that is <math> ax^2 + bx + c </math> factors into the form <math> (x + d)^2</math> for some real number d.
 
2) If <math> b^2 - 4ac = 0</math> the equation <math> ax^2 + bx + c </math> has a single real double root, that is <math> ax^2 + bx + c </math> factors into the form <math> (x + d)^2</math> for some real number d.
−
3) If <math> b^2 - 4ac < 0</math> the equation <math> ax^2 + bx + c = 0<math> has two distinct complex roots, <math> z \text{ and } \overline{z}</math>
+
 
 +
3) If <math> b^2 - 4ac < 0</math> the equation <math> ax^2 + bx + c = 0</math> has two distinct complex roots, <math> z \text{ and } \overline{z}</math>
  
 
Geometrically each of these situations tells us the following about the graph of <math> ax^2 + bx + c</math>
 
Geometrically each of these situations tells us the following about the graph of <math> ax^2 + bx + c</math>
  
 
1) In this case the graph will cross the x-axis twice.
 
1) In this case the graph will cross the x-axis twice.
 +
 
2) The graph will touch, and bounce away from the x-axis, staying on the same side. You can think of this like bouncing a tennis ball against a line on a tennis court, where the line is the x-axis. As long as you do not throw the tennis ball hard
 
2) The graph will touch, and bounce away from the x-axis, staying on the same side. You can think of this like bouncing a tennis ball against a line on a tennis court, where the line is the x-axis. As long as you do not throw the tennis ball hard
 
enough to create a crater the tennis ball has to stay "above" the line.
 
enough to create a crater the tennis ball has to stay "above" the line.
 +
 
3) The graph will not even touch the x-axis, always staying on the same side.
 
3) The graph will not even touch the x-axis, always staying on the same side.
  
 
[[Math_5|'''Return to Topics Page]]
 
[[Math_5|'''Return to Topics Page]]

Latest revision as of 12:55, 30 September 2015

Introduction

The motivation for defining complex numbers comes from wanting to define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}} in a way that would be compatible with the properties of exponents. Mathematicians accomplished this by defining a new "variable", i. Thus, we have the immediate property that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^2 = -1 } , the same way Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\sqrt{3})^2 = 3 } . Now we can define the Complex Numbers to be all numbers of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A + Bi } for real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} . The number A is called the real part of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A + Bi} , while B is the imaginary part of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A + Bi} .

Properties

Before we start describing the properties that Complex Numbers satisfy, we have to define what it means for Complex Numbers to be equal,

1) If A + Bi and C + Di are two complex numbers, then A + Bi = C + Di if and only if A = C and B = D. For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + 3i \neq 2 + 6i} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \neq 2 }

2) If A + Bi and C + Di are two complex numbers, then we can add them together as follows: (A + Bi) + (C + Di) = (A + B) + (C + D)i

3) Once again taking two complex numbers A + Bi and C + Di, we can take the difference of the two numbers, A + Bi - (C + Di) = (A - C) + (B - D)i

4) Finally we can multiply our two complex numbers together, using FOIL and the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^2 = -1 } to find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A + Bi)(C + Di) = AC + ADi + BCi + BDi^2 = (AC - BD) + (AD + BC)} For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3 + 4i)(2 - i) = 6 - 3i + 8i -4i^2 = (6 + 4) + (8 - 3)i = 10 + 5i }

Conjugate

Since the Complex Numbers have similar properties to the Real Numbers, we would like to know if Complex Numbers have multiplicative inverses, that is if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A + Bi} is a Complex Number is there another Complex Number C + Di such that (A + Bi)(C + Di) = 1. We know this holds for non-zero Real Numbers, by taking the reciprocal. For example, the multiplicative inverse of 5 is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{5}}


Before we can answer this question we are going to define the Conjugate:

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = A + Bi } is a Complex Number, then its conjugate, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{z}} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A - Bi}

Now we have the property that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = A + Bi } then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\overline{z} = A^2 + B^2}

Now we can look at some properties of the Conjugate:

Given complex numbers z and w:

1)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\overline{\overline{z}}) = z }

2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{z + w} = \overline{z} + \overline{w}}

3) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{z\cdot w} = \overline{z}\overline{w} }


Now we are ready to find the inverse of any non-zero complex number A + Bi, that is find a complex number C + Di such that (A + Bi)(C + Di) = 1.

By the properties above, we know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A + Bi)(A - Bi) = A^2 + B^2} So given any complex number A + Bi, the inverse is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A - Bi}{A^2 + B^2}} . Another way to think of this is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{A + Bi} = \frac{A - Bi}{A^2 + B^2}} .

Example: Find the inverse of 3 + 4i.

First we find the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 + B^2} for A = 3 and B = 4. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2 + B^2 = 3^2 + 4^2 = 9 + 16 = 25} . Now the inverse of 3 + 4i is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3 - 4i}{25}}


Another common question that requires knowledge of the inverse is to simplify an expression of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A + Bi}{C +Di}} .

Example:

Simplify Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ 1 + 2i}{5 + 3i}} .

From the discussion about inverses above, we can rewrite this expression as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 + 2i)\cdot \frac{1}{5 + 3i}} , and the inverse of the second term is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5 - 3i}{ 5^2 + 3^2} = \frac{5 - 3i}{34}} . Putting this all together our expression has turned into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 + 2i)\cdot \frac{5 - 3i}{34}} . To finish the problem we just need to use FOIL and simplify. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( 1 + 2i)\cdot \frac{5 - 3i}{34} = \frac{ 5 + 10i -3i -6i^2}{34} = \frac{11 + 7i}{34}}

Powers if i

Since the number i is so integral to defining the complex numbers, it is useful to know what happens when you raise i to different powers. Some powers of i are provided here:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^1 = i}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^2 = -1}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^3 = -i}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^4 = 1 } .

We can notice that each time we multiply i by itself 4 times we get 1. So if we want to know what i to any integer power is, we just have to find the remainder of the power when divided by 4 and use the powers of i that we see above.

For example, let's say we want to find the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{49} } . We divide 49 by 4 to get a reminder of 1. Looking at our values above, we can conclude that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{49} = i }

Revisiting the Quadratic Formula

On this page we learned how to solve quadratic equations using the quadratic formula. Now that we know about Complex Numbers we can classify all possible solutions to a quadratic equation and give it some geometric meaning. By geometric meaning, we can describe the behavior of the graph of the function.


We start by reminding the reader of the quadratic formula: Given a quadratic equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c = 0 } where a, b, and c are real numbers with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq 0} , the solutions are given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}} , that is the solutions are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \frac{b^2 + \sqrt{b^2-4ac}}{2a} \text{ and } x = \frac{b^2 - \sqrt{b^2-4ac}}{2a}} .

The behavior of the graph is completely determined by the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2 - 4ac} and we start by explaining how the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2 - 4ac } affects the zeros of the quadratic equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c = 0 }

1) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2 - 4ac > 0} the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c } has two distinct real roots.

2) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2 - 4ac = 0} the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c } has a single real double root, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c } factors into the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x + d)^2} for some real number d.

3) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2 - 4ac < 0} the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c = 0} has two distinct complex roots, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \text{ and } \overline{z}}

Geometrically each of these situations tells us the following about the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + bx + c}

1) In this case the graph will cross the x-axis twice.

2) The graph will touch, and bounce away from the x-axis, staying on the same side. You can think of this like bouncing a tennis ball against a line on a tennis court, where the line is the x-axis. As long as you do not throw the tennis ball hard enough to create a crater the tennis ball has to stay "above" the line.

3) The graph will not even touch the x-axis, always staying on the same side.

Return to Topics Page

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